Buffalo has the rep for snow, but, in the state, Binghamton, Rochester, Syracuse, and Albany get more snow, but Buffalo is the second largest city, so it gets the press. It's more a case of occasional snow, with one to two snowstorms per year. However, cities like Buffalo, Detroit, Cleveland, and Pittsburgh can handle virtually any amount of snow - and, ANY amount, given one to 2 days.
As far as cold, just like any other; not any worse, because Lake Erie keeps it relatively warm. It hasn't been below 0 F in years, and a real cold morning will be 8 or 10 degrees - and those aren't common.
Wisconsin, Upper Michigan, Minnesota, and the Dakotas are REALLY cold (like 30 below on cold nights), so it's a relative thing.
As a kid, I can remember 2 green Christmases in Buffalo; and, lest you think I'm biased, if anything, I'm biased against Buffalo. Even with the reasonable winter, it still is a bummer to me; this time 3 weeks ago, I was in Tahiti.
The medicine programs, I can't comment on, because there are so many of them divided up this way and that.